Superresolution Optical Fluctuation Imaging (SOFI)

ABSTRACT

Statistical analysis techniques based on auto- and cross-correlations/cumulants, of image stacks of fluctuating objects are used to improve resolution beyond the classical diffraction limit and to reduce the background. The time trajectory of every pixel in the image frame is correlated with itself and/or with the time trajectory of an adjacent pixel. The amplitude of these auto- or cross-correlations/cumulants of each pixel, at a given time lag or averaged or integrated over an interval of time lags, is used as the intensity value of that pixel in the generated superresolved optical fluctuation image.

REFERENCE TO RELATED APPLICATIONS

This application relies for priority on provisional application#61183519 of Jörg Enderlein, filed on Jun. 2, 2009 and entitled“Superresolution optical fluctuation imaging (SOFI)”

FIELD OF THE INVENTION

This invention relates generally to sub-diffraction and backgroundreduction optical (or electromagnetic) imaging methods based on a signalprocessing method of data stacks (sequence of multiple x-y images/framesas function of time i.e. a movie). The method relies on multiple-orderauto- and cross-correlation (and auto/cross cumulants) statisticalanalysis of temporal fluctuations (caused, for example, by fluorescenceblinking / intermittency) recorded in a sequence of images (a movie).The method can be applied to superresolve stationary and non-stationarysamples composed of light- emitting, absorbing or scattering blinkingobjects. Fields of use include microscopy, telescopy, medical imaging,and other forms of electromagnetic imaging.

DESCRIPTION OF RELATED ART

The publications, patents, and other reference materials referred toherein describe the background of the invention and to provideadditional detail regarding its practice are hereby incorporated.

Spatial resolution of optical imaging methods (microscopies, telescopiesetc.) are limited by the diffraction limit of light (Abbe's limit).Optical microscopies, and in particular fluorescence microscopy, permitthree-dimensional (3D) investigation of living cells, tissues and liveorganisms. However, features smaller than approximately half theemission wavelength (˜200-300 nm) for visible light cannot be resolvedin conventional far-field microscopy due to the above mentioned limit.Similarly, astronomical observations cannot resolve neighboringcelestial objects below the diffraction limit of light.

Other non-optical imaging techniques, such as electron microscopy(scanning electron microscope, transmission electron microscope,cryo-electron microscope) and scanning probe microscopy (scanningtunneling microscope, atomic force microscope), achieve molecular-levelresolution, but are not suitable for imaging features within live cells.They are limited to analyzing surfaces (membranes) or to fixed andthinly sectioned samples.

During the last decade, the optical diffraction limit has been overcomewith the introduction of several new concepts, as discussed below.

Hell and Wichmann, “Breaking the diffraction resolution limit bystimulated emission: Stimulated-emission-depletion fluorescencemicroscopy”, 1994, Optical Society of America, Optics Letters, vol. 19,pp. 780-782, Hell and Kroug, “Ground-state-depletion fluorescencemicroscopy: A concept for breaking the diffraction resolution limit”,1995, Applied Physics B, Lasers Optics, vol. 60, pp. 495-497, Klar etal. “Subdiffraction resolution in far-field fluorescence microscopy”,Jul. 15, 1999, Optical Society of America, Optics Letters, vol. 24, No.14, pp. 954-956, U.S. Pat. No. 5,731,588, U.S. Pat. No. 7,719,679, U.S.Pat. No. 7,538,893, U.S. Pat. No. 7,430,045, U.S. Pat. No. 7,253,893,and U.S. Pat. No.7,064,824 teach stimulated emission depletion (STED)microscopy and related methods, requiring special apparatuses, specialemitters (fluorophores) and relatively strong laser illumination.

Gustafsson, “Surpassing the lateral resolution limit by a factor of twousing structured illumination microscopy”, 2000, Journal of Microscopy,vol. 198, pp. 82-87, Heintzmann et al., “Saturated patterned excitationmicroscopy: A concept for optical resolution improvement”, 2002, Journalof Optical Society of America A, vol. 19, pp. 1599-1609, Gustafsson etal., (IM)-M-5: “3D wide-field light microscopy with better than 100-nmaxial resolution”, 1999, Journal of Microscopy, vol. 195, pp. 10-16, andU.S. Pat. No. 5,671,085 teach structured illumination microscopy andsaturated structured illumination microscopy (SIM / SSIM) and imageinterference microscopy (I⁵M).

Betzig et al. “Imaging intracellular fluorescent proteins at nanometerresolution”, 2006, Science, vol. 313, pp., 1642-1645, Hess et al.“Ultra-high-resolution imaging by fluorescence photoactivationlocalization microscopy”, 2006, Biophysics Journal, vol. 91, pp.4258-4272, U.S. Pat. No. 7,710,563, U.S. Pat. No. 7,626,703, U.S. Pat.No. 7,626,695, U.S. Pat. No. 7,626,694, U.S. Pat. No. 7,535,012 teachphoto-activated localization microscopy (PALM) and related methods; Rustet al., “Subdiffraction-limit imaging by stochastic opticalreconstruction microscopy (STORM)”, 2006, Nature Methods, vol. 3, pp.793-795 teach the related stochastic optical reconstruction microscopy(STORM) method.

The development of switchable fluorescent probes also triggered theemergence of new background-reducing, contrast-enhancing techniques suchas optical lock-in detection (OLID) as taught by Marriott et al.,“Optical lock-in detection imaging microscopy for contrast enhancedimaging in living cells”, 2008, Proceedings of the National Academy ofScience, USA, vol. 105, pp. 17789-17794.

STED has achieved video-rate resolution enhancement but the method isquite demanding in terms of the labeling procedures, choice of dyes, andrequires tedious alignment procedures, which are challenging. Recentlysuperresolution microscopy at 11 Hz has been demonstrated using SIM,achieving a twofold increased lateral resolution. All superresolutionmethods are capable of enhancing the resolution in 3D, but often at theexpense of major technical demands or modifications to the microscope,as taught in Shtengel et al., “Interferometric fluorescentsuper-resolution microscopy resolves 3D cellular ultrastructure”, 2009,Proceedings of the National Academy of Science, USA, vol. 106, pp.3125-3130, Juette et al., “Three-dimensional sub-100-nm resolutionfluorescence microscopy of thick samples”, 2008, Nature Methods, vol. 5,pp. 527-529, Huang et al., “Whole-cell 3D STORM reveals interactionsbetween cellular structures with nanometer-scale resolution”, 2008,Nature Methods, vol. 5, pp. 1047-1052. PALM and STORM achieve nanometerresolution, but with the trade-off of slow acquisition speed. Theacquisition of a full superresolution image usually takes minutes tohours. Lastly, even though OLID provides fast imaging with enhancedcontrast, it lacks superresolution capability.

Lidke et al. teach in “Superresolution by localization of quantum dotsusing blinking statistics”, 2005, Opics Express vol. 13, pp. 7052-7062an analysis method for simultaneously overlapping and fluctuatingemitters based on higher-order statistical analysis for separating andlocalizing the emitters. They have developed a superresolution imagingmethod which is based on Independent Component Analysis (ICA) andblinking statistics of QDs. They demonstrated that this method iscapable ofresolving QDs which are closely spaced below the diffractionlimit. As pointed out by Lidke et al. under- or over-estimating thenumber of QDs can affect the accuracy in determination of the loci ofthe (incorrect number of) emitters. As demonstrated in the currentinvention (SOFI), no such a priori knowledge of numbers of emitters isnecessary. Ventalon et al. in “Dynamic speckle illumination (DSI)microscopy with wavelet pre-filtering”, 2007, Optics Letters, vol. 32,pp.141 7-1419, 2007 discuss a statistical analysis method based onlinear addition of variances. It is based on the evaluation offluctuations in the observed signal, but ones which are induced by theexcitation light and subsequently evaluated in an analogous way to SOFI.Even though this approach yields sectioning along the optical axis, thefact that the fluctuations are not originating from independentmicroscopic (i.e. sub-diffraction sized) emitters but from diffractionlimited speckles, imposes a fundamental limit on DSI resolution, as itis diffraction limited. The similarities between SOFI and the abovementioned methods have their origin solely in the common mathematicalconcept of correlation functions. However a detailed analysis of theseapproaches reveals dramatic differences in capabilities, resultingeffects, uses, and applications.

There is therefore a great need for a superresolution andbackground-reduced imaging method that is simple, fast, low cost, has anintrinsic 3D superresolution capabilities, non-phototoxic, and is wellsuited for live cell imaging. This method should be flexible andimplementable on image stacks acquired by various widely deployed,commercial microscopes and be simple to implement and operate.

BRIEF DESCRIPTION OF THE DRAWINGS

FIGS. 1( a)-1(f) represent the steps required to generate areduced-background, superresolved image using the SOFI method.

FIG. 2 is a flowchart detailing the protocol for generating asecond-order SOFI image.

FIGS. 3( a)-3(c) represent various ways for obtaining a SOFI image basedon the second-order auto-correlation function, using different time lagconfigurations.

FIGS. 4( a)-4(c) represent various ways for obtaining a SOFI image basedon the second-order cross-correlation function, using different timelags configurations.

FIGS. 5( a)-5(c) show the principle of generating interleaving pixels,in between physical pixels, for a SOFI image (shown is an example forthe particular second-order cumulant case).

FIG. 6 is a flowchart detailing the protocol for generating interleavingpixels using cross-cumulants and correcting for the distance factor.

FIGS. 7( a)-7(d) show an example of which pixels' time trajectories canbe cross-correlated in order to obtain a fully up-sampled second-orderSOFI image.

FIG. 8 is a flowchart detailing the protocol for generating anup-sampled second-order SOFI image using second-order auto- andcross-correlations.

FIGS. 9( a)-9(b) show how to obtain the same SOFI pixel bycross-correlating various pixels' time trajectories.

FIG. 10 compares the resolution enhancement of SOFI to that of aconventional wide-filed microscopy.

SUMMARY OF THE INVENTION

In this invention, we disclose statistical analysis techniques (based onauto- and cross-correlations/cumulants) of image stacks (movies) offluctuating objects that improve resolution beyond the classicaldiffraction limit and reduce the background. The time trajectory ofevery pixel in the image frame is correlated with itself and/or with thetime trajectory of an adjacent pixel. The amplitude of these auto- orcross-correlations/cumulants of each pixel (at a given time lag oraveraged or integrated over an interval of time lags) is used as theintensity value of that pixel in the generated superresolved SOFI image.Fields of use include microscopy, telescopy, medical imaging, and otherforms of electromagnetical imaging of blinking objects.

DETAILED DESCRIPTION OF THE INVENTION

SOFI is a method for imaging a field-of-view comprizing of independentlyblinking point-like objects (FIG. 1( a)), or structures labeled withsuch objects. ‘Point-like’ refers to emitter's dimension much smallerthan the dimension of the Point Spread Function (PSF) of the imagingsystem. Each of the emitters fluctuates (or ‘blinks’) stochastically andindependently (of each other) in one or more of its optical properties,like emission, absorption or scattering. The fluctuations can beintrinsic to the objects or induced by external means, but have to beindependent and not synchronized among the objects. The fluctuations canhave their origin in a (molecular) transition, in the (molecular)orientation or in (molecular) conformation which causes some opticalproperty to change to a measurable degree. This could for example be dueto a transition between two or more energy states of a molecule, or dueto a change in an optically polarized emission, absorption, orscattering caused by a molecular dipole reorientation. It is emphasizedthat the fluctuation (or blinking) need not be of a binary nature like,for example, an ‘on’ and an ‘off’ fluorescent states. Any measurablefluctuation could be used for the generation of a SOFI image. In theexample, emitters k and j are separated by a distance shorter than thediffraction limit.

The above mentioned field-of-view (sample, or object plane) is imagedvia an optical imaging system (microscope, telescope, etc.) in parallelusing a digital camera (or another array detector) or in series(pixel-by-pixel) (FIG. 1( b)), repeatedly (frame-by-frame), over time(i.e. the data is acquired by one of a variety of time-lapse microscopymethods). The signals from emitters k and j (and all other emitters) areintrinsically convolved with the microscope point spread function (PSF)due to the diffraction of light and recorded on a sub-diffraction grid(e.g. pixels of the CCD-camera, or a scanned image) as a function oftime, This creates a series of magnified images (a movie, a stack ofimages, or frames, FIG. 1( c)) of the field-of-view that is detailed inthe dotted box of FIG. 1( a). The configuration of the imaging andrecording systems are suited and sensitive enough to detect and recordthe objects' fluctuations (blinking fluorescent emitters in the example)in time and space. The field-of-view is magnified by the imaging systemto an extent that one pixel of the recorded image corresponds to afraction of the PSF's image dimension (i.e. the PSF is sampled, forexample, by ˜4×4-10×10 pixels).

A time trajectory (or a time trace) is extracted for each pixel r_(i),=(x_(i) y_(i) z_(i)) in the recorded image stack, representing(fluorescence) intensity fluctuations as function of time t at thatposition r_(i): F(r_(i),t) (FIG. 1( d)). i is a discrete index whichenumerates all pixels of the array detector. Each pixel's timetrajectory is composed of the weighted sum of individual emitters'signals (whose PSFs are contributing to that particular pixel). Shownare time trajectories of three hypothetical pixels i−1, i, i+1, (asnoted in FIG. 1( c)) whose time trajectories are the weighted sums ofthe signals emanating from emitters k and j.

The temporal mean value of F(r_(i),t) is calculated and substracted fromthe recorded time trajectory to yield the zero-mean fluctuation signal:δF (r_(i),t) =F(r_(i),t)−

F(r_(i),t)

.

$\left( {{\langle{f(t)}\rangle}_{t} = {\frac{1}{T}{\int_{0}^{T}{{f(t)}{t}}}}} \right.$

denotes time-averaging). Subsequently, the temporal second-orderauto-correlation function G₂ (r_(i), τ₁, τ₂)=G₂(r_(i), 0, 1) =

δF(r_(i), t+0)·δF(r_(i), t+1)

, is calculated for each pixel's time trajectory, resulting a matrix ofG₂ (r_(i), 0,1) values. Without loss of generality, we set τ₁=0 andτ₂=1,0 representing the time point of the first frame of the movie and 1representing the time lag between two adjacent frames (however, any timelag values for τ_(i), and τ₂ within the recorded time range could bechosen) (FIG. 1( e) & FIG. 3( a)). A beneficial outcome of thismathematical operation is the elimination of the uncorrelatedbackground.

A SOFI image is constructed from the matrix G₂ (r,0,1) according to:

$\begin{matrix}\begin{matrix}{{S_{2}\left( r_{i} \right)} = {G_{2}\left( {r_{i},0,1} \right)}} \\{= {\langle{\delta \; {{F\left( {r_{i},{t + 0}} \right)} \cdot \delta}\; {F\left( {r_{i},{t + 1}} \right)}}\rangle}_{t}} \\{= {\sum\limits_{k}{{U^{2}\left( {r_{k} - r_{i}} \right)}ɛ_{k}^{2}{\langle{\delta \; {{s\left( {r_{k},{t + 0}} \right)} \cdot \delta}\; {s\left( {r_{k},{t + 1}} \right)}}\rangle}_{t}}}}\end{matrix} & (1)\end{matrix}$

where each pixel is assigned the value S₂ (r_(i)) (Eq. (1) and FIG. 1(f)). k extends over all objects/emitters within thesample/field-of-view. Other choices of time lags, or a value of a finiteintegral of these functions over a chosen range of time lags could beutilized. In the example, emitters k and j are now resolved in theresulted SOFI image.

While the original image (at position r_(i)) was composed of asuperposition of point-spread functions (PSFs) of the form U(r_(k)−r_(i)) for each object/emitter k, the SOFI image is composed of asuperposition of PSFs of the form U² (r_(k)−r_(i)) , scaled by abrightness term ε₂ ^(k) and a term representing the temporal fluctuationof each emitter. For an approximated three-dimensional Gaussian PSF:

$\begin{matrix}{{U(r)} = {\exp \left( {{- \frac{x^{2} + y^{2}}{2\omega_{0}^{2}}} - \frac{z^{2}}{2\omega_{0z}^{2}}} \right)}} & (2)\end{matrix}$

the functional form of the resulted SOFI-modified PSF is:

$\begin{matrix}{\mspace{79mu} {{{U^{2}(r)} = {\exp \left( {{- \frac{x^{2} + y^{2}}{2\text{?}}} - \frac{z^{2}}{2\text{?}}} \right)}}{\text{?}\text{indicates text missing or illegible when filed}}}} & (3)\end{matrix}$

where

ω₀=ω₀/√{square root over (2)} and

ω_(0z)=ω/√{square root over (2)}. As a result, the width of the PSF isreduced by a factor of √{square root over (2)}> in each of the x, y, andz directions. Often a PSF can be approximately modeled as athree-dimensional Gaussian, justifying the above expression. This holdstrue for more accurate description of the PSF such as an Airy disk, orthe actual experimentally determined PSF. The SOFI-modified PSF stillhas the form U² (r), with the resulting contraction in width.

FIG. 2 summarizes the steps of the SOFI algorithm as described inparagraphs [0025]-[0030].

Example 1: Second Order Auto-Correlation-Based Sofi With Arbitrary TimeLags

The second-order auto-correlation function G₂ (r_(i), τ₁, τ₂) forarbitrary time lags τ₁ is given by:

G₂(r_(i), τ₁, τ₂)=

δF(r_(i), t+τ₁)·δF(r_(i), t+τ₂)

(4)

A SOFI image can be generated by using the second-order auto-correlationfunction not only for time lag τ₁=0 and τ₂ =1 as described in paragraph[0028], but for any combination of time lags τ₁, τ₂ (FIG. 3( b)). Asuperposition of many such combinations will also yield a SOFI image(Eq. (5) and FIG. 3( c)):

$\begin{matrix}{{S\left( r_{i} \right)} = {\sum\limits_{\tau_{1},\tau_{2}}{G_{2}\left( {r_{i},\tau_{1},\tau_{2}} \right)}}} & (5)\end{matrix}$

Note that τ_(j) could take finite time lags values or differencesbetween time lags.

Example 2: Higher-Order Auto-Cumulant-Based Sofi

The above descriptions of the preferred embodiment and Example 1(paragraphs [0025]-[0032]) utilize second-order auto-correlations.Higher-order SOFI images could be generated by utilizing the concept ofauto-cumulants. Auto-cumulants can be derived from the auto-correlationfunctions and are identical to them for orders 1, 2 and 3 (thefirst-order correlation corresponds to the temporal mean value).Higher-order auto-correlation functions (for pixel r_(i)) are computedaccording to:

G_(n)(r_(i), τ₁, . . . , τ_(n))=

δF(r_(i),t+τ₂) . . . δF(r_(i),t+τ_(n))

(6) utilizing auto-cumulants (J. Mendel,“Tutorial on Higher-OrderStatistics (Spectra) in Signal Processing and System Theory: TheoreticalResults and Some Applications”,1991, Proceedings of IEEE, vol. 19, no.3, Equation A-1), G_(n), (r_(i),τ₁, . . . , t_(n)) could be computedaccording to Eqs. (7) below that describe the relationship betweenauto-cumulants C_(n) and auto-correlations G_(n) (only relationships upto 4th order are shown):

$\begin{matrix}{{{C_{2}\left( {r_{i},\tau_{1},\tau_{2}} \right)} = {G_{2}\left( {r_{i},\tau_{1},\tau_{2}} \right)}}{{C_{3}\left( {r_{i},\tau_{1},\tau_{2},\tau_{3}} \right)} = {G_{3}\left( {r_{i},\tau_{1},\tau_{2},\tau_{3}} \right)}}{C_{4}\left( {r_{i},\tau_{1},\tau_{2},\tau_{3},\tau_{4}} \right)} = {{G_{4}\left( {r_{i},\tau_{1},\tau_{2},\tau_{3},\tau_{4}} \right)} - {{G_{2}\left( {r_{i},\tau_{1},\tau_{2}} \right)}{G_{2}\left( {r_{i},\tau_{3},\tau_{4}} \right)}} - {{G_{2}\left( {r_{i},\tau_{1},\tau_{3}} \right)}{G_{2}\left( {r_{i},\tau_{2},\tau_{4}} \right)}} - {{G_{2}\left( {r_{i},\tau_{1},\tau_{4}} \right)}{G_{2}\left( {r_{i},\tau_{2},\tau_{3}} \right)}}}} & (7)\end{matrix}$

similarly to Eq. (5), a higher order SOFI image could be constructed byassigning the value of a superposition of higher-order cumulantsS_(n)(r_(i)) to the corresponding pixels:

$\begin{matrix}{{S_{n}\left( r_{i} \right)} = {\sum\limits_{\tau_{1},\ldots \mspace{14mu},\tau_{n}}{C_{n}\left( {r_{i},\tau_{1},\ldots \mspace{14mu},\tau_{n}} \right)}}} & (8)\end{matrix}$

and similarly to Eq. (1), this higher-order SOFI image will have theform:

$\begin{matrix}{{S\left( r_{i} \right)} = {\sum\limits_{k}{{U^{n}\left( {r_{k} - r_{i}} \right)}ɛ_{k}^{n}{w_{k}\left( {\tau_{1},\ldots \mspace{14mu},\tau_{n}} \right)}}}} & (9)\end{matrix}$

with the original PSFs raised to the n-th power U^(n)(r_(k)-r_(i)) andmultiplied by the brightness factor ε^(n) _(k) and the temporalweighting factor w_(k)(τ_(l), . . ., τ_(n)). For a 3D Gaussian PSFapproximation, the PSF's width improves by a factor of √n . For a moreaccurate approximation of the PSF (Airy disk, or the experimentallydetermined PSF), the resulting resolution improvement is determined bythe relative widths of U^(n)(r) and U(r).

Example 3: Second-Order Cross-Correlation-Based Sofi

Analogous to the second-order auto-correlation SOFI scheme (paragraphs[0025]-[0031]), one can define a spatio-temporal second-ordercross-correlations-based SOFI approach. In this case, different pixels'r_(i1) and r_(i2) time trajectories are correlated at time lags τ₁=0 andτ₂=1. The second-order cross-correlation XG₂ is then given by (see FIG.4( a)):

XG₂(r_(i1),r_(i2)|0,1)=

δF(r_(i1),t+0)·δF(r_(i2),t+1)

, (10) Whenever the original PSF is oversampled by several pixels, andthose pixels record fluctuations from near-by emitters in a correlatedmanner, a cross-correlation could be calculated. Although timetrajectories of different pixels are used to calculate thecross-correlation terms, the value of XG₂ could be assigned to aparticular SOFI image's pixel according to:

$\begin{matrix}{{{XG}_{2}\left( {r_{i\; 1},\left. r_{i\; 2} \middle| 0 \right.,1} \right)} = {{U\left( \frac{r_{i\; 1} - r_{i\; 2}}{\sqrt{2}} \right)} \cdot {\sum\limits_{k}{{{U^{2}\left( {r_{k} - \frac{r_{i\; 1} + r_{i\; 2}}{2}} \right)} \cdot ɛ_{k}^{2}}{\langle\begin{matrix}{\delta \; {{s_{k}\left( {t + 0} \right)} \cdot}} \\{\delta \; {s_{k}\left( {t + 1} \right)}}\end{matrix}\rangle}_{t}}}}} & (11)\end{matrix}$

where k extends over all emitters present in the field-of-view/sampleand r_(k) represents emitter k's positions. r_(it) and r,_(i2) representthe pixels time trajectories' positions used for the cross-correlation.The value of XG₂ is assigned to the position

$r = \frac{r_{i\; 1} + r_{i\; 2}}{2}$

(i.e. the geometric center of r_(i1) and r,_(i2)) in the final SOFIimage. The assigned SOFI's pixel intensity is given by:

$\begin{matrix}{{S_{2}(r)} = \frac{{XG}_{2}\left( {r_{i\; 1},\left. r_{i\; 2} \middle| 0 \right.,1} \right)}{U\left( \frac{r_{i\; 1} - r_{i\; 2}}{\sqrt{2}} \right)}} & (12)\end{matrix}$

By setting r_(i1)=r_(i2) and normalizing the PSF (U(0)=1) in Eq. (12)the auto-correlation result (Eq. (1)) is recovered.

Example 4: Second-Order Cross-Correlation-Based Sofi With Arbitrary TimeLags

As in Example #1, the second-order cross-correlation approach can beextended to arbitrary time lags τ_(j):

XG₂(r_(i1),r_(i2)|τ₁,τ₂)=

δF(r_(i1), t+τ₁)·δF(r_(i2), t+τ₂)

(13)

A SOFI image can be generated by using the second-ordercross-correlation function not only for time lags τ₁₌0 and τ₂=1 asdescribed in Example 3, but for any combination of time lags τ₁,τ₂(FIG.4( b)). A superposition of many such combinations will also yield a SOFIimage (Eq. (14) and FIG. 4( c)):

$\begin{matrix}{{S_{2}(r)} = {\sum\limits_{\tau_{1},\tau_{2}}\frac{{XG}_{2}\left( {r_{i\; 1},\left. r_{i\; 2} \middle| \tau_{1} \right.,\tau_{2}} \right)}{U\left( \frac{r_{i\; 1} - r_{i\; 2}}{\sqrt{2}} \right)}}} & (14)\end{matrix}$

Note that τ_(j) could take finite time lags values or differencesbetween time lags.

Example 5: Multiple Ways For Calculating Second-OrderCross-Correlation-Based Sofi

Since the location of a second-order cross-correlation-based SOFI pixelis assigned to the

$r = \frac{r_{i\; 1} + r_{i\; 2}}{2}$

location of the geometric center of the two pixels r_(i1) and r,_(i2)(FIG. 5( a)), and since other combinations of pixel pairs could have thesame geometric center (FIG. 9( a)), it would be most efficient toutilize pairs where ||r_(i1)-r,_(i2)|| is on the order of, or smallerthan, the width of the PSF. The SOFI image would then take the form:

$\begin{matrix}{{S_{2}(r)} = {\sum\limits_{{r_{i\; 1} + r_{i\; 2}} = {2r}}\frac{{XG}_{2}\left( {r_{i\; 1},\left. r_{i\; 2} \middle| \tau_{1} \right.,\tau_{2}} \right)}{U\left( \frac{r_{i\; 1} - r_{i\; 2}}{\sqrt{2}} \right)}}} & (15)\end{matrix}$

Example 6: Higher-Order Cross-Cumulant-Based Sofi

Analogous to the higher-order temporal auto-cumulant approach (Example2), it is possible to define a higher-order spatio-temporalcross-cumulants SOFI approach, in which the time trajectories ofdifferent pixels are cross-correlated. Cross-cumulants can be derivedfrom cross-correlation functions and are identical to them for orders 2and 3. Higher-order spatio-temporal cross-correlation functions XG_(n)have as inputs n-tupel pixels' time trajectories (r_(i1), r_(i2), . . .r_(in)) and n-tupel time lags (τ₁, τ₂, . . . τ_(n)). XG_(n) is given by:

XG_(n)(r_(i1), . . . , r_(in)|τ₁, . . . , τ_(n))

δF(r_(i1), t+τ₂) . . . δF(r_(in),t+τ_(n))

(16) and could be computed utilizing cross-cumulants (J. Mendel,“Tutorial on Higher-Order Statistics (Spectra) in Signal Processing andSystem Theory: Theoretical Results and Some Applications”, 1991,Proceedings of IEEE, vol. 19, no. 3, Equation A-1).

XG_(n)(r_(i1), r_(in), . . . ,τ_(n)) could be computed according to Eqs(17) below that describe the relationship between cross-cumulants XC_(n)and cross-correlations XG_(n) (only relationships up to the fourth-orderare shown):

$\begin{matrix}{{{{XC}_{2}\left( {r_{i\; 1},r_{i\; 2},\tau_{1},\tau_{2}} \right)} = {{XG}_{2}\left( {r_{i\; 1},r_{i\; 2},\tau_{1},\tau_{2}} \right)}}{{{XC}_{3}\left( {r_{i\; 1},r_{i\; 2},r_{i\; 3},\tau_{1},\tau_{2},\tau_{3}} \right)} = {{XG}_{3}\left( {r_{i\; 1},r_{i\; 2},r_{i\; 3},\tau_{1},\tau_{2},\tau_{3}} \right)}}{{{XC}_{4}\left( {r_{i\; 1},r_{i\; 2},r_{i\; 3},r_{i\; 4},\tau_{1},\tau_{2},\tau_{3},\tau_{4}} \right)} = {{{XG}_{4}\left( {r_{i\; 1},r_{i\; 2},r_{i\; 3},r_{i\; 4},\tau_{1},\tau_{2},\tau_{3},\tau_{4}} \right)} - {{{XG}_{2}\left( {r_{i\; 1},r_{i\; 2},\tau_{1},\tau_{2}} \right)}{{XG}_{2}\left( {r_{i\; 3},r_{i\; 4},\tau_{3},\tau_{4}} \right)}} - {{{XG}_{2}\left( {r_{i\; 1},r_{i\; 3},\tau_{1},\tau_{3}} \right)}{{XG}_{2}\left( {r_{i2},r_{i\; 4},\tau_{2},\tau_{4}} \right)}} - {{{XG}_{2}\left( {r_{i\; 1},r_{i\; 4},\tau_{1},\tau_{4}} \right)}{{XG}_{2}\left( {r_{i\; 2},r_{i\; 3},\tau_{2},\tau_{3}} \right)}}}}} & (17)\end{matrix}$

Whenever the image of the original emitters' PSF is large enough to beoversampled by several pixels, and those pixels can detect correlatedfluctuations, a cross-correlation can be performed between pixels whichmutually oversample a region. Each cross-cumulant calculated can be usedto generate a SOFI image. Even though the cross-cumulants featuredifferent pixels' time trajectories as an input, it is possible toassign a unique location (pixel) for the resulted cross-cumulant valuein the final SOFI image. XC_(n) is given by:

$\begin{matrix}{{{XC}_{n}\left( {r_{i\; 1},\left. {\ldots \mspace{14mu} r_{i\; n}} \middle| \tau_{1} \right.,\ldots \mspace{14mu},\tau_{n}} \right)} = {\prod\limits_{l < j}^{n}\; {{U\left( \frac{r_{il} - r_{ij}}{\sqrt{n}} \right)} \cdot {\sum\limits_{k}{{U^{n}\left( {r_{k} - {\frac{1}{n}{\sum\limits_{m = 1}^{n}r_{im}}}} \right)} \cdot ɛ_{k}^{n} \cdot {w_{k}\left( {\tau_{1},{\ldots \mspace{14mu} \tau_{n}}} \right)}}}}}} & (18)\end{matrix}$

The location of the corresponding SOFT pixel is given by:

$\begin{matrix}{r = {\frac{1}{n}{\sum\limits_{m = 1}^{n}r_{m}}}} & (19)\end{matrix}$

(as can be seen from Eq. (18)). The higher order cross-cumulants SOFIimage is given by:

$\begin{matrix}{{S_{n}(r)} = \frac{{XC}_{n}\left( {r_{i\; 1},\left. {\ldots \mspace{14mu} r_{i\; n}} \middle| \tau_{1} \right.,\ldots \mspace{14mu},\tau_{n}} \right)}{\prod\limits_{l \leq j}^{n}\; {U\left( \frac{r_{il} - r_{ij}}{\sqrt{n}} \right)}}} & (20)\end{matrix}$

Analogous to Eq. (14) a superposition of many time lags combinations canbe used to calculate the higher order cross-cumulants SOFI image:

$\begin{matrix}{{S_{n}(r)} = {\sum\limits_{\tau_{1},\ldots \mspace{14mu},\tau_{n}}\frac{{XC}_{n}\left( {r_{i\; 1},\left. {\ldots \mspace{14mu} r_{i\; n}} \middle| \tau_{1} \right.,\ldots \mspace{14mu},\tau_{n}} \right)}{\prod\limits_{l < j}^{n}\; {U\left( \frac{r_{il} - r_{ij}}{\sqrt{n}} \right)}}}} & (21)\end{matrix}$

Since the location of the n^(th)-order cross-cumulant-based SOFI pixelis assigned to the location of

$r = {\frac{1}{n}{\sum\limits_{m = 1}^{n}r_{im}}}$

the geometric center of the n pixels and since other combinations ofpixels tupel could have the same geometric center, it would be mostefficient to utilize tupel where ||r_(il)-r_(if)|| (for all combinations(l,j)) is on the order of, or smaller than, the width of the PSF. TheSOFI image would then take the form:

$\begin{matrix}{{S_{n}(r)} = {\sum\limits_{{r_{i\; 1} + \ldots + r_{i\; n}} = {2r}}\frac{{XC}_{n}\left( {r_{i\; 1},\left. {\ldots \mspace{14mu} r_{i\; n}} \middle| \tau_{1} \right.,\ldots \mspace{14mu},\tau_{n}} \right)}{\prod\limits_{l < j}^{n}\; {U\left( \frac{r_{il} - r_{ij}}{\sqrt{n}} \right)}}}} & (22)\end{matrix}$

The resulting higher-order SOFI image contains a combination of PSFs ofthe form U^(n)(r_(k)-r), with a brightness factor e^(n) _(k) and atemporal weighting factor w_(k)(τ₁, . . . , τ_(n)). For a 3D Gaussian

PSF approximation, the PSF's width improves by a factor of √n. For amore accurate approximation of the PSF (Airy disk, or the experimentallydetermined PSF), the resulting resolution improvement is determined bythe relative widths of U^(n)(r) and U(r).

Example 7: Generating Interleaved Pixels By Cross-Cumulants

Interleaving pixels could be generated by cross-correlating trajectoriesof pixels whose geometric center falls in between physical ('real')pixels. FIG. 5( a) shows how to generate an interleaved pixel bysecond-order cross-correlation. Two adjacent pixels' time trajectoriesare cross-correlated, resulting in an interleaved pixel located inbetween the two physical pixels. For example, a second-ordercross-correlation between a pixel time trajectory at coordinate r₁ =100and a pixel time trajectory at coordinate r₂ =101 produces a pixel atcoordinate r =100.5. This approach holds true also for higher ordercumulants. The cross-cumulant is weighted by a factor which is dependendon the distances of the pixel time trajectory used for the generation ofthe cross-cumulant (FIGS. 5( b) & 5(c)) :

$\begin{matrix}{\prod\limits_{l < j}^{n}\; {U\left( \frac{r_{il} - r_{ij}}{\sqrt{n}} \right)}} & (23)\end{matrix}$

This distance factor (Eq. (23)) has to be known for each pixel in theSOFI image. Then the SOFI value for all pixels can be calculatedaccording to Eq. (20), resp. Eq. (21) resp. Eq (22) (see also FIG. 5(c)). Considering the possible combinations of neighboring pixels, asecond-order cross-correlation calculation between neighboring pixelscreates effective pixels halfway between each horizontal, vertical, anddiagonal pairing (FIG. 6 and FIGS. 7( a)-7(d)). Likewise, the higherorder cross-correlations produce even larger numbers of effective pixelsgiven by each possible pairing. For example the fourth-ordercross-cumulant allows the pairing of 4 different pixels' timetrajectories.

In contrast to interpolation, these cross-cumulant derived interleavedpixels provide true increased resolution. For example, a pair ofemitters which is spaced closer than the pixel sampling size (andtherefore not resolved in the original, conventional image) can beresolved by using cross-correlations/cross-cumulants to produce asufficient number of SOFI pixels so that an intensity dip in between twointensity peaks (representing the two emitters) becomes visible. As aresult, by using the cross-correlation approach in combination with theauto-correlation approach across multiple cumulant orders, it ispossible to create continually increasing numbers of pixels in theresulting image, allowing a single image stack to produce SOFI imageswith a range of resolutions, a range of numbers of resulting pixels, andresolution exceeding even the original pixel sampling size. In general,the cross-correlation approach has the advantage that even though thePSF is shrinking in size, the sampling frequency which has to be used torecord the SOFI image (e.g. the effective pixel size of the camera:nm/pixel) does not have to be adjusted. The original sampling frequencywith respect to the PSF could be maintained. A flowchart of thegeneration of such up sampled SOFI images can be found in FIG. 8,demonstrating this second order technique.

Example 8: Signal Enhancement By Cross-Cltmulants

Cross-cumulants can be used to calculate the SOFI value of a pixel(virtual or real) multiple times using different pixel-time trajectorypairs (respective pixel-time trajectory triples or n-tupels for then^(th)-order SOFI image). This approach can be used to enhance a noisysignal, since each cross-cumulant will carry the same information butobtained from different pixel-time trajectory pairs (respective pixeltime trajectory-triples or n-tupels for the n^(th) order SOFI image).

Which pixel time trajectory has to be cross- correlated can be seen fromEq. 19. For example, see FIGS. 9( a) and 9(b).

Example 9: Microscopy Applications

SOFI is particularly suited for fluorescence microscopy applications. Itcan be used to produce superresolution and background reduction onalmost any fluorescence microscope and with any fluorophore whichindependently transitions between two or more intensity, lifetime,polarization, or spectral states. Examples of such fluorophoretransitions include quantum dot blinking, fluorophore triplet statesduring which they do not emit, and photoswitchable probes which have aprobability of turning off or changing spectra under a specificwavelength (and possibly low intensity) illumination.

The SOFI approach is notable in comparison to other superresolutiontechniques since it works on a wide variety of standard fluorescencemicroscopes with no need for modification to the instrument. This ispossible because the key to the resolution enhancement is thestochastically independent blinking/fluctuations of the emitters. Themicroscope must only provide the means to record an image stack of theseemitters so that their blinking/fluctuations could be analyzed. As aresult, SOFI can be performed on widefield microscopes where a camera isused to image an entire field of view simultaneously. It can also beused on modified widefeld microscopes such as a confocal spinning discmicroscope, where sectioning and elimination of out of focus light isimproved by a spinning disk of pinholes, and on Total InternalReflection Fluorescence (TIRF) microscopes, where internal reflection atthe coverslip is used to select a very narrow illumination slice (closeto the glass surface), causing a very narrow PSF in the depth (z)direction. In each case, the microscope provides the initial(diffraction-limited)

PSF, and subsequent utilization of the SOFI algorithm providessuperresolution in all three dimensions as well as background reduction.

The SOFI approach can also be used on a raster scanning setup in which asingle excitation spot is scanned through the sample (scanning beam orscanning stage) to produce an image. This can be accomplished by simplyscanning slow enough so that fluctuations could be observed at a singlepoint. Alternatively one can try to sample the fluctuations by scanningthe sample multiple times, so quickly that the beam returns to the samepoint fast enough to over-sample fluctuations with respect to thefluctuation rate (i.e. fast enough so that the signal is stillsufficiently correlated in time). Even a third option could be used: thecombination τ_(j) =0 for all j (see also [0047]). For this approach thetemporal correlation can be lost completely. However, it will not bepossible to apply the cross-cumulant approach anymore.

Example 10: Non-Fluorescent Microscopy Applications

SOFI can be performed on all kinds of objects that blink/fluctuate intheir electromagnetic emission, absorption or scattering properties. Forexample, a stochastically reorienting / rotationaly diffusing small goldnano-rod nanoparticle will scatter light anisotropically. A wide-,dark-field microscope equipped with polarization optics and a suitablecamera could image and record fluctuations in light scattering of suchreorienting objects and acquire a SOFI-compatible data set. Similarly,changes in absorption dipole orientation could be exploited andsubjected to SOFI analysis.

Example 11: Non-Microscopy Applications

SOFI applications are not limited to microscopy. Any electromagneticfar-field imaging system or wave phenomena that is subjected to thediffraction limit, that records signals from fluctuating point-likeemitting/absorbing/scattering sources could take advantage of the SOFIalgorithm. Possible applications include telescopy, medical imaging, andother forms of electromagnetical imaging.

Example 12: Selection Of Time-Lags, Blinking Timescale, Shot Noise,Frame Rate, And Image Stack Duration

The correlation function works for any arbitrary time lag τ_(j), orrelative time delays between signal values which are correlated.However, a careful matching between the typical fluctuation/blinkingtimescale (rate), the image acquisition frame rate, and the chosencorrelation time lag is necessary for correct implementation of the SOFIalgorithm. It also impacts the inclusion or removal of usually unwantedshort timescale detector fluctuations such as afterpulsing, the tendencyin some detectors to produce counts shortly after other counts, or shotnoise, which is the uncorrelated statistical fluctuation in signalintensity from one frame to the immediately following one.

The simplest selection of time lags is to set all τ_(j)=0 , which makesthe correlation functions equivalent to the mean-centered moments, andthe generalized cumulants equal to the more conventional cumulants. Forexample, under this special case the second-order SOFI image isequivalent to the temporal variance of each pixel. This is conceptuallysimplest, and it permits the observation of very short timescalefluctuations; however, it results in the inclusion of shot noisebehavior which reduces the ability to resolve independently fluctuatingemitters. The shot noise can be removed after the fact if it obeys aknown behavior, such as the measured noise distribution of a camera, orthe Poisson-distributed shot noise of a photon counting detector.Removal of the shot noise in this manner allows the sensitivity to shorttimescale fluctuations of the emitters while still obtainingsuperresolution.

However, a SOFI image can be generated which is intrinsically shot-noisefree, if one or more non-zero time lags are chosen. This approach isguaranteed to suppress the shot-noise contribution regardless of itsactual statistical distribution. Once non-zero time lags are used, it isnecessary to consider the timescale of the intrinsic blinking behaviorexhibited by the emitters. For obtaining good sensitivity in observingblinking behavior with this approach, the typical blinking timescleshould be long enough so that one blinking period ('off time) usuallypersists across more than one integration time per frame (the inverse ofthe frame rate) of the original image stack.

When cross-correlations between independent pixels is used (instead ofauto-correlations), shot noise fluctuations are suppressed even for zerotime lag. Thus, by using either a subtraction of the shot-noisedistribution (when using auto-correlations) or a cross-correlationapproach with zero time lag, it is possible to look at blinkingtimescales as short as the acquisition time of a single frame, whichcould be shorter than the inverse of the frame rate, as for example, byusing strobed excitation (whereby the illumination which excites thefluorescent emitters is only turned on for a brief time during eachframe, allowing the generation of SOFI images from blinking timescalesmuch shorter than the frame rate). This approach could, for example, beused to image short timescale triplet state behavior where a fluorophoreblinks by entering a non-emitting triplet state for a duration ofmicroseconds to milliseconds.

The other critical aspect of selecting a timescale for analysis is forthe prevention of slow timescale drifts in the mean (as for example, dueto slow thermal or mechanical drifts of the sample stage) disrupting thegeneration of a SOFI image. This is resolved by either generating thecorrelation functions in segments much shorter than the drift time-scaleso that the mean is centered for each segment, or by adding uncorrelatedrandom noise to normalize the means. For example, fluorescence sampleswith dyes commonly experience bleaching, where the fluorophores have acertain probability with each excitation of permanently entering a darkstate. This results in the mean intensity exhibiting an exponentialdecay where, depending on the fluorophore and excitation power, thedecay time can range from seconds to many minutes or longer. Since allregions of the image with that dye will bleach similarly, this makesmany emitters, which for SOFI should be independent, effectivelycorrelated with each other in that they will transition from a brighterstate at the beginning of the image stack toward a dark state at the endof the image stack. By splitting the image stack into a set of shorterstacks, or using the random noise solution, this problem could bemitigated.

Example 13: Stacks Of Image Stacks—Generating Sofi Movies

The speed of acquiring a SOFI image is given by the imaging system andthe timescale of the fluctuations. Therefore one can acquire multipleimage stacks and generate SOFI image sequences (SOFI movies) featuringsuperresolution. Specifically, when the object which is imaged ismoving, the potentially short acquisition times prove advantageous,because otherwise a blurred superresolution image could result. Thecalculation of SOFI images can be done very efficiently and quicklyusing either software or hardware-implemented approaches.

Example 14: Background Reduction

The generation of a SOFI image relies on the analysis of temporal signalfluctuations relative to the mean signal level. As a result, backgroundsignals which remain constant or which produce very little fluctuationare suppressed, resulting in background-reduced (and sometimes evenbackground-free) and contrast-enhanced SOFI images. This backgroundreduction can remove very large constant signals, allowing a muchsmaller fluctuating signal to become visible. This can permit theobservation of structures (superresolved or larger) which wouldotherwise be lost or invisible in a large background.

When a time-lag selection greater than zero is chosen, the contrastenhancement selects only signals which persist in a pixel across morethan one frame of the original image stack. As a result, shot noise, aspart of the background, is reduced, as are many other signals which aremuch shorter than the duration of the frame. Cross-correlation alsoreduces any fluctuating background which is present only in a singlepixel, for example detector afterpulsing, cosmic rays, dark counts, andshot noise even for zero time lag, and it selects for contrastenhancement only the fluctuations which happen in a correlated manneracross multiple pixels.

In the case of cellular imaging with fluorescence, a common problemwhich disrupts image quality is the high degree of background providedby both out of focus light and cell autofluorescence (where the naturalcontents of the cell emit some light in response to the excitationlight). As both out of focus light and cell autofluorescence arenon-fluctuating signals, they are eliminated in the SOFI image,producing images of substantially higher quality and solving animportant contrast problem in cellular imaging.

Out of focus light and scattering significantly contribute to reducedcontrast and reduced imaging performance in live animals or tissueimaging, The same principle described above for cellular imaging applieshere too; the background reduction and contrast enhancement of SOFI cangreatly aid such in-vivo applications.

Example 15: 3-Dimensional (3D) Imaging

The SOFI approach intrinsically shrinks the PSF in all 3 dimensions. Itis therefore possible to achieve three-dimensional superresolution. Infact, even a two-dimensional SOFI image has increased resolution in thez dimension since it enhances the signals coming from emitters that arecloser to the focal plane. A 3D superresolved stack is obtained bysimply acquiring image stacks at each of several different depths(sections), and calculating a SOFI image for each section, resulting ina 3D superresolved and background reduced SOFI image stack.

Interleaving pixels along the z dimension can be calculated as inExample #7 by cross-correlation signals originating from differentsections or by interpolating the image stacks to intermediate sectionsprior to implementing the SOFI algorithm. However, this introducesrequirements for the timescale of imaging. To obtain intermediatepixels, the frames at each depth must be measured either simultaneouslyor fast enough so that the correlation between frames of differentdepths is preserved.

To obtain a 3D SOFI image with a cross-correlation between pixels atdifferent depths, one would need to measure the signal at two depthseither simultaneously or in tandem, but fast enough relative to thefluctuations time scale (so that the correlation is preserved).

REDUCTION TO PRACTICE

Dertinger T et al., “Fast, background-free, 3D super-resolution opticalfluctuation imaging (SOFI)”, 2009, Proceedings of the National Academyof Sciences, vol. 106, pp. 22287- 22292 give a detailed description ofSOFI implementation and reduction to practice. Below we give a briefsummary: microtubules of 3T3 fibroblast cells were immuno-stained withquantum dots (QDs), imaged in a wide-field microscope with a CCD camera,and the data stack was analysed by the SOFI algorithm. The resultedcross-section of the superresolved and background-reduced SOFI image(FIGS. E-H of above mentioned reference) is shown in FIG. 10. Listedbelow are the details for obtaining the SOFI image.

Cell labeling_(L) NIH-3T3 (ATCC, Manassas, VA, USA) cells were grown upto a confluence level of ˜80% in Dulbecco's Modified Eagle's Medium(ATCC, Catalog No. 30-2002) plus 10% fetal calf serum (10082-147,Invitrogen, Carlsbad) plus 100 units penicillin- streptomycin(Pen-Strep, 15140122, Invitrogen, Carlsbad). For fixation the followingprocedure has been applied. Cells were incubated at RT for 15 min withCB buffer (10mM MES, pH6.2, 140 mM NaCl, 2.5 mM EGTA , 5 mM MgC1₂), 11%Sucrose, 3.7% paraformaldehyde, 0.5% glutaraldehyde, 0.25% Triton as afixative. Quenching was done with 0.5mg/ml sodium borohydride in CB for8 min. Cells were washed once with PBS and blocked in 2% BSA+PBS for 1hour at RT. Microtubules were stained using 1:500 dilution of DM1Aanti-a-tubulin monoclonal Antibody (Sigma Inc.) in 2% BSA+PBS. Cellswere then washed 3 times with PBS and incubated for 1h at RT with a 1:400 dilution of quantum dots (QDs) QD625 labeled goat F(ab)₂, anti-mouseIgG Antibodies (H+L) (Invitrogen Inc., Carlsbad) in 6% BSA +PBS. Cellswere washed 3 times with PBS. All steps were performed in a humiditychamber. Specimens were dehydrated by floating the coverslipssequentially for 5 seconds in 30%, 70%, 90% and 100% ethanol.Subsequently they were gently spin-coated (˜500 rpm) with 1mg/ml PVA.

Microscope Setup: Movies were taken on an inverted wide-field microscope(Olympus, IX71). A 470 nm LED array device was used as a light source(Lumencor Inc., Aura Light Engine, Beaverton, Oregon, USA). Sampleexcitation and fluorescence collection was done using a high numericalaperture objective (Olympus, UPlanApo 60x, 1.45, Oil, Center Valley, PA,USA). Excitation light was filtered from fluorescence using a 620/40bandpass emission filter (D620/40, Chroma Technology Corp, Rockingham,VT, USA). The fluorescence light was focused on a CCD camera (Andor,iXon^(EM)+885, Belfast, Northern Ireland). Magnification was adjusted toobtain 35 nm / pixel.

Data acquisition: A movie was acquired with 3000 frames, 100 ms/frame.

Data analysis: Movies were analyzed using the SOFI algorithm describedin paragraphs [0025]- [0031] above using a custom written Matlabsoftware.

FIG. 10. Compares the resolution enhancement of SOFI. Intensity profilesextracted from the dotted lines in FIG. 5 E-H of Dertinger T et al. Thesolid line indicates a cross-section in the original wide-field image.The dashed line indicates the same cross-section in the second-orderSOFI image. This comparison clearly establishes gain in resolution andreduction in background.

1. A method for analyzing a field-of-view of independently blinkingobjects, comprising the steps of: i) selecting a sample or anobservation image comprised of objects from the class of opticallysignaling objects that stochastically and independently fluctuate; ii)acquiring a sequence of repeated optical images of the object as afunction of time, producing an x, y, t image stack of pixels; iii)extracting a time trajectory for each pixel from said image stack; iv)calculating an autocorrelation or autocummulant of each pixel's timetrajectory to at least the second order, and v) generating asuperresolved and background-reduced image from the calculated functionscomprising pixels whose intensity value is the amplitude, with a chosentime lag, of the autocorrelation or autocummulant function, or the valueof a finite integral of the autocorrelation or autocummulant functionover a range of time lags.
 2. A method as set forth in claim 1 above,wherein the independently blinking objects could be separated bydistances shorter than the diffraction limit -resolved distance of theimaging system used.
 3. A method as set forth in claim 1 above, whereinthe sample objects comprising the sample or the observation image areselected from the class comprising emitting, absorbing or scatteringmaterials and objects labeled with such materials.
 4. A method as setforth in claim 1 above, wherein the sequence of optical images isproduced from a scanning beam or scan sample, in parallel or inseries-wide field, or in pixel-by-pixel sequence.
 5. A method as setforth in claim 1 above, wherein the generated functions comprise eithercross-correlation or cross-cummulant functions.
 6. A method as set forthin claim 5 above, wherein the correlation or cummulant functions are ofhigher order than the second order.
 7. A method as set forth in claim 6above, wherein the acquired sequence of optical images comprisesmultiple sets of image stacks.
 8. A method for imaging the field-of-viewof objects whose resolution is below the diffraction limit, in a systemin which optical emitters in the field-of-view could be spaced apart bydistances shorter than the diffraction limit-resolved distance of theimaging system used and fluctuate independently with time, the methodcomprising the steps of: (i) optically imaging the field-of-view with anoptical system of a given point spread function as a function of time toproduce x, y, t stacks, each stack comprising the variant pixelsequences whose intensities are superposition of signals emanating fromthe independent emitters of sub-diffraction size; (ii) extracting a timetrajectory from said image stacks for each pixel therein; (iii)calculating a correlation or cumulant function of the time trajectory ofeach pixel to at least a second order; (iv) generating a new imagecomprising pixels, and wherein pixels'intensity values are related tothe correlation or cumulant function in accordance with a chosen timelag, or a finite integral of said functions over a range of time lags,to derive a superresolution, background-reduced optical image.
 9. Amethod as set forth in claim 8 above, wherein the emitters' dimensionsare substantially smaller than the PSF, and wherein the method includesmagnifying the field-of-view such that pixels are a fraction of the PSFimage dimension, and a zero-mean fluctuation signal is derived andsubtracted from the time trajectory of each pixel and a second-ordercorrelation or cumulant function is calculated for each pixel toconstruct a matrix defining the intensity of SOFI pixels in a secondorder SOFI image.